Is there any way to apply linear transformations on a vector other than matrix multiplication?
You can sometimes exploit the structure of your matrix to perform faster matrix multiplication. For example, if your matrix is sparse (or dense), there are algorithms that exploit this fact. 

In your case, you can actually compute $A^n$ in less time than $\mathcal{O}(n^3$). For example, have a look at this question at CS SE and this one at Stack Overflow (SO). Note that the provided solutions may not be numerically stable, so I am not suggesting you use them.

Moreover, if you perform your operations on the GPU, they could be faster in practice. See e.g. this question at SO and this one at SciComp SE.